POP247
Models for St Lucia - Wi-Fi & BCC traffic
Radoslaw Panczak
25 June, 2022
1 Vehicle counts
1.1 Original data
Detailed example counts for week of 2018-12-03 across
type and direction.
Overview of months, inbound
1.2 Stocks on campus
1.2.1 Without modification
1.2.2 With midnight ‘reset’
Frequency of net from whole period
Frequency of stocks from whole period
2 Wi-Fi
2.1 Original data
Detailed example counts for week of 2018-12-09 across
type of data.
Overview of months, connected users
2.2 Distribution of counts
Frequency of counts from whole period
2.3 Distribution fits
3 Time series comparisons
3.1 Inbound, connected users
3.2 Outbound, connected users
3.3 Net, connected users
3.4 Stocks, connected users
4 Correlations, connected users
5 Models
5.1 INLA setup
# ####
# Jeffreys prior
a1 <- 5e-5
b1 <- 5e-5
lgprior1 <- list(prec = list(param = c(a1, b1)))
# Gelman prior
a2 <- -0.5
b2 <- 5e-5
lgprior2 <- list(prec = list(param = c(a2, b2)))
# iid prior
# Schrödle & Held 2010 & Blangiardo et al 2013
a0 <- 1
b0 <- 0.1
prior.nu <- list(prec = list(param = c(a0, b0)))
# intercept & fixed
inla.set.control.fixed.default()
# intercept ~ N(0,0)
# other fixed effects ~ N(0, 0.001)
#
# where the format is N(mean, precision)
# precision = inverse of the variance.
# PC prior
U <- 1
hyper.prec = list(theta = list(
prior = "pc.prec",
param = c(U, 0.01)
))
# scaling
inla.setOption(scale.model.default = TRUE)5.2 RW1
5.3 One trend for stocks
mod1 <- inla(users_mac ~
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.704, Running = 4.94, Post = 0.535, Total = 6.18
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.416 0.006 7.404 7.416 7.427 7.416 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 6.93 0.136 6.67
Precision for stock 3.36 0.708 2.19
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 6.93 7.20 6.93
Precision for stock 3.28 4.96 3.12
Deviance Information Criterion (DIC) ...............: 79444.34
Deviance Information Criterion (DIC, saturated) ....: 153614.32
Effective number of parameters .....................: 121.68
Watanabe-Akaike information criterion (WAIC) ...: 79467.01
Effective number of parameters .................: 138.42
Marginal log-Likelihood: -47532.98
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.4 One trend for stocks + weekday
mod2 <- inla(users_mac ~
weekday +
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.603, Running = 6.01, Post = 0.503, Total = 7.12
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.373 0.014 7.345 7.373 7.400 7.373 0
weekdayMonday 0.030 0.020 -0.008 0.030 0.069 0.030 0
weekdaySaturday 0.000 0.021 -0.041 0.000 0.041 0.000 0
weekdaySunday 0.026 0.021 -0.016 0.026 0.068 0.026 0
weekdayThursday 0.232 0.020 0.194 0.232 0.271 0.232 0
weekdayTuesday -0.041 0.020 -0.079 -0.041 -0.002 -0.041 0
weekdayWednesday 0.030 0.020 -0.009 0.030 0.068 0.030 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 7.22 0.140 6.94
Precision for stock 3.95 0.788 2.63
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 7.21 7.50 7.21
Precision for stock 3.87 5.71 3.72
Deviance Information Criterion (DIC) ...............: 79219.74
Deviance Information Criterion (DIC, saturated) ....: 153389.73
Effective number of parameters .....................: 120.52
Watanabe-Akaike information criterion (WAIC) ...: 79230.99
Effective number of parameters .................: 127.12
Marginal log-Likelihood: -47456.16
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.5 One trend for stocks + weekend + weekday
mod3 <- inla(users_mac ~
weekend +
weekday +
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.5, Running = 6.45, Post = 0.288, Total = 7.24
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.373 0.014 7.345 7.373 7.400 7.373 0
weekendWeekend 0.009 18.257 -35.837 0.008 35.824 0.009 0
weekdayMonday 0.030 0.020 -0.008 0.030 0.069 0.030 0
weekdaySaturday -0.009 18.257 -35.854 -0.009 35.807 -0.009 0
weekdaySunday 0.017 18.257 -35.828 0.017 35.833 0.017 0
weekdayThursday 0.232 0.020 0.194 0.232 0.271 0.232 0
weekdayTuesday -0.041 0.020 -0.079 -0.041 -0.002 -0.041 0
weekdayWednesday 0.030 0.020 -0.009 0.030 0.068 0.030 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 7.22 0.141 6.94
Precision for stock 3.93 0.797 2.63
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 7.22 7.50 7.21
Precision for stock 3.84 5.76 3.65
Deviance Information Criterion (DIC) ...............: 79219.89
Deviance Information Criterion (DIC, saturated) ....: 153389.87
Effective number of parameters .....................: 120.60
Watanabe-Akaike information criterion (WAIC) ...: 79230.98
Effective number of parameters .................: 127.05
Marginal log-Likelihood: -47456.69
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.6 Comparing
5.6.1 DIC
5.6.2 WAIC
